You need to buy his share of the house off him/he needs to sell his share of the house to you. But you need the price to be at or at least near "market value" - otherwise his creditors may try to void the sale if it was done to prevent other creditors getting their fair share of what they are owed. He can't just 'sign it over to you' - his creditors will have an interest in the property.
If a legitimate sale is concluded then you have sole ownership of the house - though probably secured after remortgaging. The money your partner got from you will go towards paying off his debts.
If the security on the house remains as it is, you could find yourself with a half-ownership of an almost unsellable asset.
See a solicitor to arrange the necessary registration and property transfer arrangements.